Solving a Logistic Model - Population Data

[botty_12]

Hey

Im in the middle of modelling a logistic model off of population data but I am having a little bit of trouble. I am using a three parameter model
y=M/(1+Ce^-kt) and have set up three different equations to solve simultaneously. I originally used the value for when t=0 eliminating the k in one equation but I'm not too sure if i can use that to sub into the other equations. As far as solving it, I get to a point then become lost. Any help guys?

5.3= M/(1+Ce^-10k)

62.9=M/(1+Ce^-100k)

226.5=M/(1+Ce^-190k)

3.9 = M/(1+C)

 

[HallsofIvy]

Hey

Im in the middle of modelling a logistic model off of population data but I am having a little bit of trouble. I am using a three parameter model
y=M/(1+Ce^-kt) and have set up three different equations to solve simultaneously. I originally used the value for when t=0 eliminating the k in one equation but I'm not too sure if i can use that to sub into the other equations. As far as solving it, I get to a point then become lost. Any help guys?

5.3= M/(1+Ce^-10k)

62.9=M/(1+Ce^-100k)

226.5=M/(1+Ce^-190k)

3.9 = M/(1+C)[/QUOTE]

These are the equations you get when you substite t= 10, 100, 190, and 0, right? How did you get the y values on the left? You have 4 equations in 3 unknowns. If you are not absolutely certain that the function is of the given form, then you might not be able to find values of k, M and C that satisfy all 4. One thing you can do now is divide one equation by another, eliminating M.
Dividing the second equation by the first, 62.9/5.3= 11.87= (1+ Ce^-10k)/(1+ Ce^-100k) so 1+ Ce^-10k= 11.87+ 11.87Ce^-1o0k or
C(e^-10k- 11.87e^-100k)= 10.87.

Dividing the third equation by the first, 226.5/5.3= 42.75= (1+ Ce^-10k)/(1+ Ce^-190k) so 42.75- 42.75Ce^-190k= 1+ Ce^-10k or C(e^-10k- 42.75e^-190k)= 41.75.

Dividing one of those equations by the other eliminates C leaving a single equation in k.

 

[tacman]

Hi there,

It looks like you are on the right track with your logistic model and equations. Solving a logistic model can be tricky, but there are a few steps you can take to make it easier.

First, it's important to understand the variables in your model. In your equation y=M/(1+Ce^-kt), M represents the maximum population, C represents the initial growth rate, and k represents the carrying capacity. These values can be determined from your data.

Next, to solve the equations simultaneously, you can use substitution. Since you have three equations and three unknowns (M, C, and k), you can solve for one variable in terms of the other two in one equation, then substitute that into the other equations.

For example, you can solve the first equation for M by multiplying both sides by (1+Ce^-10k) and then dividing by 5.3. This will give you M=5.3(1+Ce^-10k). You can then substitute this value for M into the other two equations and solve for C and k.

Alternatively, you can use a graphing calculator or a software like Excel to plot your data and use the trendline function to find the best fit for your logistic model. This can also help you determine the values for M, C, and k.

Overall, it may take some trial and error to find the best fit for your data, but with some persistence and using the right tools, you should be able to solve your logistic model. Good luck!